In two subsequent parts monotonicity and comparison results will be studied for arbitrary dynamic systems governed by nonnegative matrices. In the present Part I, it is shown that monotonicity and comparison results, as kown for Markov reward chains, do carry over rather smoothly to the general nonnegative case for marginal, total and average reward structures. These results are not only of theoretical interest by themselves, but also essential for the more practical continuous-time case in Part II.